常用坐标系及其变换

121.2.3.4.5.6.12¾¾“”dddOXYZOdOYdOXdOZOdOXdOXdOYY111OXYZO1OY1OX1OZ1OX1OX1OY222OXYZO2OY2OX2OZ2OX2OX2OY333OXYZO3OY3OX3OZ3OX3OX3OYOOOYOXOZOXOXYrθϕOPθrϕOPOXYOPOXYOXPlllOXYZOlOYlOXlOZlOXlOXlOYlOX1.2.θVψdOXdOX2OX′Vγ3OXVVVVγθψγθψ⎫⇒⎬⎭αβ1dd1OXOXYOXϑ1dd1dd1OXOXYOXOXOXOXψ′′11OYOXγ111OXYZ333OXYZdddOXYZ222OXYZαβθVψVγϑψγ222OXYZdddOXYZVVVddVVVdZYXZZYXYZYXXψψθψθθθψψθψθcossinsinsincos0cossinsincossincoscos222222222⋅+⋅⋅+⋅⋅−=⋅+⋅+⋅=⋅+⋅⋅−⋅⋅=''22''212''22cos0sin010sin0cosdVVdVVdXXXYYCYZZZψψψψ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⎣⎦'222'2222'222cossin0sincos0001XXXYYCYZZZθθθθ⎡⎤⎡⎤⎡⎤−⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦12'22'12122'22222cos0sincossin0010sincos0sin0cos001coscossincossinsincos0cossinsinsindddVVVVVVVVXXXYCYCCYZZZXYZψψθθθθψψθψθψψθθθψθψ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎡⎤−⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⋅−⋅=−⋅⋅222cosVVXYZψ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦21⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡333222cossin0sincos0001ZYXZYXVVVVγγγγ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅⋅−⋅−⋅=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅⋅−⋅−⋅=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡333222cossin0sincos0001cossinsinsincos0cossinsincossincoscoscossinsinsincos0cossinsincossincoscosZYXZYXZYXVVVVVVVVVVVVVVVVdddγγγγψψθψθθθψψθψθψψθψθθθψψθψθ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅+⋅⋅−⋅+⋅⋅⋅−⋅−⋅⋅+⋅⋅⋅+⋅⋅−⋅=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡333coscossinsinsinsincoscossinsinsincossincoscoscossincossinsincossinsinsincoscossincoscosZYXZYXVVVVVVVVVVVVVVVVVVVVdddγψγψθγψγψθψθγθγθθγψγψθγψγψθψθ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⋅+⋅⋅−⋅+⋅⋅⋅−⋅−⋅⋅+⋅⋅⋅+⋅⋅−⋅=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡111coscossinsinsinsincoscossinsinsincossincoscoscossincossinsincossinsinsincoscossincoscosZYXZYXdddγψγψϑγψγψϑψϑγϑγϑϑγψγψϑγψγψϑψϑ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡333111cos0sinsinsincoscossinsincossincoscosZYXZYXβββααβαβααβα23323132331000cossin0sincosVVVVXXXYYCYZZZγγγγ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=−=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦''311''3121''113cos0sin010sin0cosXXXYYCYZZZββββ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦⎣⎦'111'1131'111cossin0sincos0001XXXYYCYZZZαααα⎡⎤⎡⎤⎡⎤−⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦'2311'2131211231'2311XXXXYCYCCYCCCYZZZZ⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥===⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦212112coscossincossinsincoscossinsincoscossinsinsincossinsinsincossincoscossinsinsincoscoscosVVVVVVVVVVXXYYZZαβαββαγαβγαγαβγβγαγαβγαγαβγβγ⋅−⋅⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=⋅+⋅⋅−⋅−⋅⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⋅−⋅⋅+⋅⎣⎦⎣⎦⎣⎦21:11TTAAXAYYAXAX−−====